A new definition of analytic adjoint ideal sheaves via the residue functions of log-canonical measures I

A new definition of analytic adjoint ideal sheaves for quasi-plurisubharmonic (quasi-psh) functions with only neat analytic singularities is studied and shown to admit some residue short exact sequences which are obtained by restricting sections of the newly defined adjoint ideal sheaves to some unions of $\sigma$-log-canonical ($\sigma$-lc) centres. The newly defined adjoint ideal sheaves induce naturally some residue $L^2$ norms on the unions of $\sigma$-lc centres which are invariant under log-resolutions. They can also describe unions of $\sigma$-lc centres without the need of log-resolutions even if the quasi-psh functions in question are not in a simple-normal-crossing configuration. This is hinting their potential use in discussing the $\sigma$-lc centres even when the quasi-psh functions in question have more general singularities. Furthermore, their relations between the algebraic adjoint ideal sheaves of Ein--Lazarsfeld as well as those of Hacon--McKernan are described in order to illustrate their role as a (potentially finer) measurement of singularities in the minimal model program. In the course of the study, a local $L^2$ extension theorem is proven, which shows that holomorphic sections on any unions of $\sigma$-lc centres can be extended holomorphically to some neighbourhood of the unions of $\sigma$-lc centres with some $L^2$ estimates. The proof does not rely on the techniques in the Ohsawa--Takegoshi-type $L^2$ extension theorems.